Sliding - UPC
Kalman Filters and Adaptive Windows for Learning in Data Streams Albert Bifet
Ricard Gavaldà
Universitat Politècnica de Catalunya
DS’06 Barcelona
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
1 / 29
Outline 1
Introduction
2
The Kalman Filter and the CUSUM Test
3
The ADWIN Algorithm
4
General Framework
5
K-ADWIN
6
Experimental Validation of K-ADWIN
7
Conclusions
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
2 / 29
Introduction
Introduction
Data Streams Sequence potentially infinite High amount of data: sublinear space
High Speed of arrival: small constant time per example
Estimation and prediction Distribution and concept drift K-ADWIN : Combination Kalman filter ADWIN : Adaptive window of recently seen data items.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
3 / 29
Introduction
Introduction
Problem Given an input sequence x1 , x2 , . . . , xt , . . . we want to output at instant t a prediction xbt+1 minimizing prediction error: |xbt+1 − xt+1 | considering distribution changes overtime.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
4 / 29
Introduction
Time Change Detectors and Predictors: A General Framework Estimation -
xt -
Estimator
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
5 / 29
Introduction
Time Change Detectors and Predictors: A General Framework Estimation -
xt -
Estimator
Alarm -
A. Bifet, R. Gavaldà (UPC)
Change Detect.
-
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
5 / 29
Introduction
Time Change Detectors and Predictors: A General Framework Estimation -
xt -
Estimator
Alarm -
Change Detect.
-
6 6 ? -
A. Bifet, R. Gavaldà (UPC)
Memory
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
5 / 29
Introduction
Introduction
Our generic proposal: Use change detector Use memory
Our particular proposal: K-ADWIN Kalman filter as estimator Use ADWIN as change detector with memory [BG06]
Application Estimate statistics from data streams In Data Mining Algorithms based on counters, replace them for estimators.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
6 / 29
Introduction
Data Mining Algorithms with Concept Drift No Concept Drift
Concept drift
DM Algorithm output
input -
Counter5
-
input
DM Algorithm
output
-
-
Static Model
Counter4 Counter3
6
Counter2 Counter1
A. Bifet, R. Gavaldà (UPC)
-
Change Detect.
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
7 / 29
Introduction
Data Mining Algorithms with Concept Drift No Concept Drift
Concept Drift
DM Algorithm output
input -
DM Algorithm
Counter5
-
output
input -
Estimator5
Counter4
Estimator4
Counter3
Estimator3
Counter2
Estimator2
Counter1
Estimator1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
-
7 / 29
The Kalman Filter and the CUSUM Test
The Kalman Filter Optimal recursive algorithm Minimum mean-square error estimator Estimate the state x ∈ h then alarm and gt = 0
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
10 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 1 W1 = 01010110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 10 W1 = 1010110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 101 W1 = 010110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 1010 W1 = 10110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 10101 W1 = 0110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 101010 W1 = 110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 1010101 W1 = 10111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 10101011 W1 = 0111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 |ˆ µW0 − µ ˆW1 | ≥ c : CHANGE DETECTED! W0 = 101010110 W1 = 111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 Drop elements from the tail of W W0 = 101010110 W1 = 111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 01010110111111 Drop elements from the tail of W W0 = 101010110 W1 = 111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1 01010110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10 1010110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 101 010110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1010 10110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10101 0110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 101010 110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1010101 10111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10101011 0111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 101010110 111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1010101101 11111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10101011011 1111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 101010110111 111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1010101101111 11
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models
W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10101011011111 1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06]
ADWIN has rigorous guarantees On ratio of false positives On ratio of false negatives On the relation of the size of the current window and change rates
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
13 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Theorem At every time step we have: 1
(Few false positives guarantee) If µt remains constant within W , the probability that ADWIN shrinks the window at this step is at most δ.
2
(Few false negatives guarantee) If for any partition W in two parts W0 W1 (where W1 contains the most recent items) we have |µW0 − µW1 | > , and if s ≥4·
3 max{µW0 , µW1 } 4n ln min{n0 , n1 } δ
then with probability 1 − δ ADWIN shrinks W to W1 , or shorter. A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
14 / 29
The ADWIN Algorithm
Data Streams Algorithm ADWIN2 [BG06] ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Sliding Window Model 1010101 101 11 1 1 Content:
4
2
2
1 1
Capacity:
7
3
2
1 1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
15 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Insert new Item 1010101 101 11 1 1 Content:
4
2
2
1 1
Capacity:
7
3
2
1 1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
16 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Insert new Item 1010101 101 11 1 1
1
Content:
4
2
2
1 1
1
Capacity:
7
3
2
1 1
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
16 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Compressing Buckets 1010101 101 11 1 1
1
Content:
4
2
2
1 1
1
Capacity:
7
3
2
1 1
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
16 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Compressing Buckets 1010101 101 11 11 1 Content:
4
2
2
2
1
Capacity:
7
3
2
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
17 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Compressing Buckets 1010101 101 11 11 1 Content:
4
2
2
2
1
Capacity:
7
3
2
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
17 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Compressing Buckets 1010101 10111 11 1 Content:
4
4
2
1
Capacity:
7
5
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
18 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Detecting Change: Delete last Bucket 1010101 10111 11 1 Content:
4
4
2
1
Capacity:
7
5
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
19 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Detecting Change: Delete last Bucket 10111 11 1 Content:
4
2
1
Capacity:
5
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
19 / 29
General Framework
General Framework Time Change Detectors and Predictors : Type I Example (Kalman Filter) Estimation -
xt -
Estimator
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
20 / 29
General Framework
General Framework Time Change Detectors and Predictors : Type II Example (Kalman Filter + CUSUM) Estimation -
xt -
Estimator
Alarm -
A. Bifet, R. Gavaldà (UPC)
Change Detect.
-
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
20 / 29
General Framework
General Framework Time Change Detectors and Predictors : Type III Example (Adaptive Kalman Filter) Estimation -
xt -
Estimator 6
-
A. Bifet, R. Gavaldà (UPC)
Memory
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
20 / 29
General Framework
General Framework Time Change Detectors and Predictors : Type IV Example (ADWIN, Kalman Filter+ADWIN) Estimation -
xt -
Estimator
Alarm -
Change Detect.
-
6 6 ? -
A. Bifet, R. Gavaldà (UPC)
Memory
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
20 / 29
General Framework
Time Change Detectors and Predictors: A General Framework
No memory
Memory
No Change Detector
Type I Kalman Filter
Type III Adaptive Kalman Filter
Change Detector
Type II Kalman Filter + CUSUM
Type IV ADWIN Kalman Filter + ADWIN
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
21 / 29
General Framework
Time Change Detectors and Predictors: A General Framework
No memory
Memory
No Change Detector
Type I Kalman Filter
Type III Adaptive Kalman Filter Q,R estimated from window
Change Detector
Type II Kalman Filter + CUSUM
Type IV ADWIN Kalman Filter + ADWIN Q,R estimated from window
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
21 / 29
K-ADWIN
K-ADWIN = ADWIN + Kalman Filtering Estimation -
xt -
Kalman
Alarm -
ADWIN
-
6 6 ? -
ADWIN Memory
R = W 2 /50 and Q = 200/W , where W is the length of the window maintained by ADWIN.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
22 / 29
Experimental Validation of K-ADWIN
Tracking Experiments
KALMAN: R=1000;Q=1 Error= 854.97
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
23 / 29
Experimental Validation of K-ADWIN
Tracking Experiments ADWIN : Error= 674.66
KALMAN: R=1000;Q=1 Error= 854.97
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
23 / 29
Experimental Validation of K-ADWIN
Tracking Experiments K-ADWIN Error= 530.13
ADWIN : Error= 674.66
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
23 / 29
Experimental Validation of K-ADWIN
Naïve Bayes Example Data set that describes the weather conditions for playing some game.
outlook sunny sunny overcast rainy rainy rainy overcast
temp. hot hot hot mild cool cool cool
humidity high high high high normal normal normal
windy false true false false false true true
play no no yes yes yes no yes
Assume we have to classify the following new instance: outlook temp. humidity windy play sunny cool high true ? A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
24 / 29
Experimental Validation of K-ADWIN
Naïve Bayes Assume we have to classify the following new instance: outlook temp. humidity windy play sunny cool high true ? We classify the new instance: νNB = arg
max ν∈{yes,no}
P(νj )P(sunny |νj )P(cool|νj )P(high|νj )P(true|νj )
Conditional probabilities can be estimated directly as frequencies: P(ai |νj ) =
number of instances with attribute ai and class νj total number of training instances with class νj
Create one estimator for each frequence that needs estimation A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
24 / 29
Experimental Validation of K-ADWIN
Experimental Validation of K-ADWIN
We test Naïve Bayes Predictor and k-means clustering Method: replace counters by estimators Synthetic data where change is controllable Naïve Bayes: We compare accuracy of Static model: Training of 1000 samples every instant Dynamic model: replace probabilities counters by estimators
computing the ratio
A. Bifet, R. Gavaldà (UPC)
%Dynamic Static
with tests using 2000 samples.
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
25 / 29
Experimental Validation of K-ADWIN
Naïve Bayes Predictor
ADWIN Kalman Q = 1, R = 1000 Kalman Q = 1, R = 1 Kalman Q = .25, R = .25 Adaptive Kalman CUSUM Kalman K-ADWIN Fixed-sized Window Fixed-sized Window Fixed-sized Window Fixed-sized Window
A. Bifet, R. Gavaldà (UPC)
Width
%Static
%Dynamic
% Dynamic/Static
32 128 512 2048
83,36% 83,22% 83,21% 83,26% 83,24% 83,30% 83,24% 83,28% 83,30% 83,28% 83,24%
80,30% 71,13% 56,91% 56,91% 76,21% 50,65% 81,39% 67,64% 75,40% 80,47% 82,19%
96,33% 85,48% 68,39% 68,35% 91,56% 60,81% 97,77% 81,22% 90,52% 96,62% 98,73%
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
26 / 29
Experimental Validation of K-ADWIN
k -means Clustering
ADWIN Kalman Q = 1, R = 1000 Kalman Q = 1, R = 100 Kalman Q = .25, R = .25 Adaptive Kalman CUSUM Kalman K-ADWIN Fixed-sized Window Fixed-sized Window Fixed-sized Window Fixed-sized Window Fixed-sized Window Fixed-sized Window
A. Bifet, R. Gavaldà (UPC)
Width
σ = 0.15 Static Dynamic
32 128 512 2048 8192 32768
9,72 9,72 9,71 9,71 9,72 9,72 9,72 9,72 9,72 9,72 9,72 9,72 9,72
21,54 19,72 17,60 22,63 18,98 18,29 17,30 25,70 36,42 38,75 39,64 43,39 53,82
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
27 / 29
Experimental Validation of K-ADWIN
Results
No estimator ever does much better than K-ADWIN K-ADWIN does much better than every other estimators in at least one context. Tracking problem K-ADWIN and ADWIN automatically do about as well as the Kalman filter with the best set of fixed covariance parameters. Naïve Bayes and k -means: K-ADWIN does somewhat better than ADWIN and far better than any memoryless Kalman filter.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
28 / 29
Conclusions
Conclusions and Future Work
K-ADWIN tunes itself to the data stream at hand, with no need for the user to hardwire or precompute parameters. Better results than either memoryless Kalman Filtering or sliding windows with linear estimators. Future work : Tests on real-world, not only synthetic data. Other learning algorithms: algorithms for induction of decision trees.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
29 / 29
Ricard Gavaldà
Universitat Politècnica de Catalunya
DS’06 Barcelona
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
1 / 29
Outline 1
Introduction
2
The Kalman Filter and the CUSUM Test
3
The ADWIN Algorithm
4
General Framework
5
K-ADWIN
6
Experimental Validation of K-ADWIN
7
Conclusions
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
2 / 29
Introduction
Introduction
Data Streams Sequence potentially infinite High amount of data: sublinear space
High Speed of arrival: small constant time per example
Estimation and prediction Distribution and concept drift K-ADWIN : Combination Kalman filter ADWIN : Adaptive window of recently seen data items.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
3 / 29
Introduction
Introduction
Problem Given an input sequence x1 , x2 , . . . , xt , . . . we want to output at instant t a prediction xbt+1 minimizing prediction error: |xbt+1 − xt+1 | considering distribution changes overtime.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
4 / 29
Introduction
Time Change Detectors and Predictors: A General Framework Estimation -
xt -
Estimator
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
5 / 29
Introduction
Time Change Detectors and Predictors: A General Framework Estimation -
xt -
Estimator
Alarm -
A. Bifet, R. Gavaldà (UPC)
Change Detect.
-
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
5 / 29
Introduction
Time Change Detectors and Predictors: A General Framework Estimation -
xt -
Estimator
Alarm -
Change Detect.
-
6 6 ? -
A. Bifet, R. Gavaldà (UPC)
Memory
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
5 / 29
Introduction
Introduction
Our generic proposal: Use change detector Use memory
Our particular proposal: K-ADWIN Kalman filter as estimator Use ADWIN as change detector with memory [BG06]
Application Estimate statistics from data streams In Data Mining Algorithms based on counters, replace them for estimators.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
6 / 29
Introduction
Data Mining Algorithms with Concept Drift No Concept Drift
Concept drift
DM Algorithm output
input -
Counter5
-
input
DM Algorithm
output
-
-
Static Model
Counter4 Counter3
6
Counter2 Counter1
A. Bifet, R. Gavaldà (UPC)
-
Change Detect.
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
7 / 29
Introduction
Data Mining Algorithms with Concept Drift No Concept Drift
Concept Drift
DM Algorithm output
input -
DM Algorithm
Counter5
-
output
input -
Estimator5
Counter4
Estimator4
Counter3
Estimator3
Counter2
Estimator2
Counter1
Estimator1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
-
7 / 29
The Kalman Filter and the CUSUM Test
The Kalman Filter Optimal recursive algorithm Minimum mean-square error estimator Estimate the state x ∈ h then alarm and gt = 0
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
10 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 1 W1 = 01010110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 10 W1 = 1010110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 101 W1 = 010110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 1010 W1 = 10110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 10101 W1 = 0110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 101010 W1 = 110111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 1010101 W1 = 10111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 W0 = 10101011 W1 = 0111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 |ˆ µW0 − µ ˆW1 | ≥ c : CHANGE DETECTED! W0 = 101010110 W1 = 111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 101010110111111 Drop elements from the tail of W W0 = 101010110 W1 = 111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Example W = 01010110111111 Drop elements from the tail of W W0 = 101010110 W1 = 111111
ADWIN: A DAPTIVE W INDOWING A LGORITHM 1 Initialize Window W 2 for each t > 0 3 do W ← W ∪ {xt } (i.e., add xt to the head of W ) 4 repeat Drop elements from the tail of W 5 until |ˆ µW0 − µ ˆW1 | ≥ c holds 6 for every split of W into W = W0 · W1 7 Output µ ˆW
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
11 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1 01010110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10 1010110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 101 010110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1010 10110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10101 0110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 101010 110111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1010101 10111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10101011 0111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 101010110 111111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1010101101 11111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10101011011 1111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 101010110111 111
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 1010101101111 11
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Window Management Models
W = 101010110111111 Equal & fixed size subwindows 1010 1011011 1111 D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. 2004
Total window against subwindow 10101011011 1111 J. Gama, P. Medas, G. Castillo, and P. Rodrigues. Learning with drift detection. 2004
ADWIN: All Adjacent subwindows 10101011011111 1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
12 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06]
ADWIN has rigorous guarantees On ratio of false positives On ratio of false negatives On the relation of the size of the current window and change rates
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
13 / 29
The ADWIN Algorithm
Algorithm ADWIN [BG06] Theorem At every time step we have: 1
(Few false positives guarantee) If µt remains constant within W , the probability that ADWIN shrinks the window at this step is at most δ.
2
(Few false negatives guarantee) If for any partition W in two parts W0 W1 (where W1 contains the most recent items) we have |µW0 − µW1 | > , and if s ≥4·
3 max{µW0 , µW1 } 4n ln min{n0 , n1 } δ
then with probability 1 − δ ADWIN shrinks W to W1 , or shorter. A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
14 / 29
The ADWIN Algorithm
Data Streams Algorithm ADWIN2 [BG06] ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Sliding Window Model 1010101 101 11 1 1 Content:
4
2
2
1 1
Capacity:
7
3
2
1 1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
15 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Insert new Item 1010101 101 11 1 1 Content:
4
2
2
1 1
Capacity:
7
3
2
1 1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
16 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Insert new Item 1010101 101 11 1 1
1
Content:
4
2
2
1 1
1
Capacity:
7
3
2
1 1
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
16 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Compressing Buckets 1010101 101 11 1 1
1
Content:
4
2
2
1 1
1
Capacity:
7
3
2
1 1
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
16 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Compressing Buckets 1010101 101 11 11 1 Content:
4
2
2
2
1
Capacity:
7
3
2
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
17 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Compressing Buckets 1010101 101 11 11 1 Content:
4
2
2
2
1
Capacity:
7
3
2
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
17 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Compressing Buckets 1010101 10111 11 1 Content:
4
4
2
1
Capacity:
7
5
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
18 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Detecting Change: Delete last Bucket 1010101 10111 11 1 Content:
4
4
2
1
Capacity:
7
5
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
19 / 29
The ADWIN Algorithm
Algorithm ADWIN2 ADWIN2 using a Data Stream Sliding Window Model, can provide the exact counts of 1’s in O(1) time per point. tries O(log W ) cutpoints uses O( 1 log W ) memory words the processing time per example is O(log W ) (amortized) and O(log2 W ) (worst-case). Detecting Change: Delete last Bucket 10111 11 1 Content:
4
2
1
Capacity:
5
2
1
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
19 / 29
General Framework
General Framework Time Change Detectors and Predictors : Type I Example (Kalman Filter) Estimation -
xt -
Estimator
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
20 / 29
General Framework
General Framework Time Change Detectors and Predictors : Type II Example (Kalman Filter + CUSUM) Estimation -
xt -
Estimator
Alarm -
A. Bifet, R. Gavaldà (UPC)
Change Detect.
-
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
20 / 29
General Framework
General Framework Time Change Detectors and Predictors : Type III Example (Adaptive Kalman Filter) Estimation -
xt -
Estimator 6
-
A. Bifet, R. Gavaldà (UPC)
Memory
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
20 / 29
General Framework
General Framework Time Change Detectors and Predictors : Type IV Example (ADWIN, Kalman Filter+ADWIN) Estimation -
xt -
Estimator
Alarm -
Change Detect.
-
6 6 ? -
A. Bifet, R. Gavaldà (UPC)
Memory
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
20 / 29
General Framework
Time Change Detectors and Predictors: A General Framework
No memory
Memory
No Change Detector
Type I Kalman Filter
Type III Adaptive Kalman Filter
Change Detector
Type II Kalman Filter + CUSUM
Type IV ADWIN Kalman Filter + ADWIN
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
21 / 29
General Framework
Time Change Detectors and Predictors: A General Framework
No memory
Memory
No Change Detector
Type I Kalman Filter
Type III Adaptive Kalman Filter Q,R estimated from window
Change Detector
Type II Kalman Filter + CUSUM
Type IV ADWIN Kalman Filter + ADWIN Q,R estimated from window
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
21 / 29
K-ADWIN
K-ADWIN = ADWIN + Kalman Filtering Estimation -
xt -
Kalman
Alarm -
ADWIN
-
6 6 ? -
ADWIN Memory
R = W 2 /50 and Q = 200/W , where W is the length of the window maintained by ADWIN.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
22 / 29
Experimental Validation of K-ADWIN
Tracking Experiments
KALMAN: R=1000;Q=1 Error= 854.97
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
23 / 29
Experimental Validation of K-ADWIN
Tracking Experiments ADWIN : Error= 674.66
KALMAN: R=1000;Q=1 Error= 854.97
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
23 / 29
Experimental Validation of K-ADWIN
Tracking Experiments K-ADWIN Error= 530.13
ADWIN : Error= 674.66
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
23 / 29
Experimental Validation of K-ADWIN
Naïve Bayes Example Data set that describes the weather conditions for playing some game.
outlook sunny sunny overcast rainy rainy rainy overcast
temp. hot hot hot mild cool cool cool
humidity high high high high normal normal normal
windy false true false false false true true
play no no yes yes yes no yes
Assume we have to classify the following new instance: outlook temp. humidity windy play sunny cool high true ? A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
24 / 29
Experimental Validation of K-ADWIN
Naïve Bayes Assume we have to classify the following new instance: outlook temp. humidity windy play sunny cool high true ? We classify the new instance: νNB = arg
max ν∈{yes,no}
P(νj )P(sunny |νj )P(cool|νj )P(high|νj )P(true|νj )
Conditional probabilities can be estimated directly as frequencies: P(ai |νj ) =
number of instances with attribute ai and class νj total number of training instances with class νj
Create one estimator for each frequence that needs estimation A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
24 / 29
Experimental Validation of K-ADWIN
Experimental Validation of K-ADWIN
We test Naïve Bayes Predictor and k-means clustering Method: replace counters by estimators Synthetic data where change is controllable Naïve Bayes: We compare accuracy of Static model: Training of 1000 samples every instant Dynamic model: replace probabilities counters by estimators
computing the ratio
A. Bifet, R. Gavaldà (UPC)
%Dynamic Static
with tests using 2000 samples.
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
25 / 29
Experimental Validation of K-ADWIN
Naïve Bayes Predictor
ADWIN Kalman Q = 1, R = 1000 Kalman Q = 1, R = 1 Kalman Q = .25, R = .25 Adaptive Kalman CUSUM Kalman K-ADWIN Fixed-sized Window Fixed-sized Window Fixed-sized Window Fixed-sized Window
A. Bifet, R. Gavaldà (UPC)
Width
%Static
%Dynamic
% Dynamic/Static
32 128 512 2048
83,36% 83,22% 83,21% 83,26% 83,24% 83,30% 83,24% 83,28% 83,30% 83,28% 83,24%
80,30% 71,13% 56,91% 56,91% 76,21% 50,65% 81,39% 67,64% 75,40% 80,47% 82,19%
96,33% 85,48% 68,39% 68,35% 91,56% 60,81% 97,77% 81,22% 90,52% 96,62% 98,73%
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
26 / 29
Experimental Validation of K-ADWIN
k -means Clustering
ADWIN Kalman Q = 1, R = 1000 Kalman Q = 1, R = 100 Kalman Q = .25, R = .25 Adaptive Kalman CUSUM Kalman K-ADWIN Fixed-sized Window Fixed-sized Window Fixed-sized Window Fixed-sized Window Fixed-sized Window Fixed-sized Window
A. Bifet, R. Gavaldà (UPC)
Width
σ = 0.15 Static Dynamic
32 128 512 2048 8192 32768
9,72 9,72 9,71 9,71 9,72 9,72 9,72 9,72 9,72 9,72 9,72 9,72 9,72
21,54 19,72 17,60 22,63 18,98 18,29 17,30 25,70 36,42 38,75 39,64 43,39 53,82
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
27 / 29
Experimental Validation of K-ADWIN
Results
No estimator ever does much better than K-ADWIN K-ADWIN does much better than every other estimators in at least one context. Tracking problem K-ADWIN and ADWIN automatically do about as well as the Kalman filter with the best set of fixed covariance parameters. Naïve Bayes and k -means: K-ADWIN does somewhat better than ADWIN and far better than any memoryless Kalman filter.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
28 / 29
Conclusions
Conclusions and Future Work
K-ADWIN tunes itself to the data stream at hand, with no need for the user to hardwire or precompute parameters. Better results than either memoryless Kalman Filtering or sliding windows with linear estimators. Future work : Tests on real-world, not only synthetic data. Other learning algorithms: algorithms for induction of decision trees.
A. Bifet, R. Gavaldà (UPC)
Kalman Filters and Adaptive Windows for Learning in Data Streams DS’06 Barcelona
29 / 29
